Simulations of supersymmetric Yang-Mills theory
on the lattice
Georg Bergner
ITP WWU Münster
Sign Workshop: Sept 22, 2012
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
1
Supersymmetric Yang-Mills theory on the lattice
2
The sign problem in supersymmetric Yang-Mills theory
3
Summary of the Results
4
Conclusions
In collaboration with I. Montvay, G. Münster, U. D. Özugurel,
S. Piemonte, D. Sandbrink
2/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Supersymmetry
SUSY
fermions bosons;
n
o
Q , Q̄ ∼ γ µ Pµ
⇒ strict paring of states; except ground state
E3
E2
E3
E2
E1
E1
E0
E3
E2
E3
E2
E1
E1
E0
E0 6= 0
E0 = 0
fermion
boson
∆ can be 6= 0
unbroken SUSY
fermion
boson
∆ = 0; all states paired
spontaneous SUSY breaking
Witten index1 :
∆ = nBE =0 − nFE =0 = Tr(−1)F = limβ→0 Tr(−1)F exp(−βH)
1
[Witten, Nucl.Phys.B202 (1982)]
3/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Super Yang-Mills theory
Supersymmetric Yang-Mills theory:
1
i
mg
/
L = Tr − Fµν F µν + ψ̄ Dψ−
ψ̄ψ
4
2
2
supersymmetric counterpart of Yang-Mills theory;
but in several respects similar to QCD
ψ Majorana fermion in the adjoint representation
gluino mass term mg ⇒ soft SUSY breaking
4/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Lattice supersymmetry
contradiction: locality = lattice SUSY1
no Ginsparg-Wilson solution (so far)2
⇒ fine tuning problem
low dimensions: fine tuning/locality problem solved3
SYM theory: tuning possible
1
2
3
[Kato, Sakamoto, So, JHEP 0805 (2008)], [GB, JHEP 1001 (2010)]
[GB, Bruckmann, Pawlowski, Phys. Rev. D 79 (2009)]
[Golterman,Petcher Nucl. Phys. B319 (1989)],
[Catterall, Gregory, Phys. Lett. B 487 (2000)],
[Giedt, Koniuk, Poppitz, Yavin, JHEP 0412 (2004)],
[G.B, Kästner, Uhlmann, Wipf, Annals Phys. 323 (2008)],
[Baumgartner, Wenger, PoS LATTICE 2011],. . .
5/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Supersymmetric Yang-Mills theory on the lattice
Lattice action:
SL = β
X
P
1−
1
<UP
Nc
+
1X
λ̄x (Dw (mg ))xy λy
2 xy
“brute force” discretization: Wilson fermions
Dw = 1 − κ
4 h
X
µ=1
(1 − γµ )α,β Tµ + (1 + γµ )α,β Tµ†
Tµ λ(x) = Vµ λ(x + µ̂);
κ=
i
1
2(mg + 4)
links in adjoint representation: (Vµ )ab = 2Tr[Uµ† T a Uµ T b ]
explicit breaking of symmetries: chiral Sym. (UR (1)), SUSY
6/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Recovering symmetry
Ward identities of supersymmetry and chiral symmetry:
tuning of κ(mg ) to recover chiral symmetry 1
same tuning to recover supersymmetry 2
Fine-tuning:
chiral limit = SUSY limit +O(a), obtained at critical κ
good realization: overlap/domainwall fermions (but too
expensive)3
practical determination of critical κ:
limit of zero mass of adjoint pion (a − π)
⇒ definition of gluino mass: ∝ (ma−π )2
1
2
3
[Bochicchio et al., Nucl.Phys.B262 (1985)]
[Veneziano, Curci, Nucl.Phys.B292 (1987)]
[Fleming, Kogut, Vranas, Phys. Rev. D 64 (2001)], [Endres, Phys. Rev. D 79 (2009)],
[JLQCD, PoS Lattice 2011]
7/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
The sign problem in supersymmetric theories
Z ∝ ∆ (periodic boundary conditions)
∆ = 0 from fluctuating sign of fermion path integral
Majorana fermions:
Z
R
√
1
Dψe − 2 ψ̄Dψ = Pf(CD) = sign(Pf(CD)) det D
⇒ severe sign problem if spontaneous SUSY breaking possible1
1
[Wozar, Wipf, Annals Phys. 327 (2012)], [Wenger]
8/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
The Sign problem in SYM and on the lattice
continuum SU(N) supersymmetric Yang-Mills theory: ∆ = N
⇒ no sign problem in the continuum
Wilson fermions: sign problem even in SYM
reweighting: sign(Pf(CD))
general lattice SUSY: modification of fermion path integral by
Wilson term requires special concern
9/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Sign problem and eigenvalues
γ5 -Hermiticity: γ5 D γ5 = D † ⇒ pairing λ, λ∗
charge conjugation: C D C T = DT
⇒ degenerate eigenvalues λ1 = λN/2+1
Q
⇒ det(D) = N/2 λ2i positive
p
QN/2
| Pf(C (D −σ1l))| = det(D −σ1l) = i=1 |λi − σ|
Pfaffian polynomial in σ
N/2
⇒
Pf(C D) =
Y
λi
i=1
number of negative paired real eigenvalues of D even / odd
⇒ positive / negative Pfaffian
10/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Sign problem and the eigenvalues
contribution of neg. signs: reduced in continuum limit;
enlarged in chiral limit
methods: next talk
further applications: determinant sign in Nf = 1 QCD
further applications: spectral decomposition, index
11/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Status of the simulations
main focus: mass-spectrum of SYM
simulations similar to Nf = 1 QCD
PHMC: approximate | Pf(CD)|
improvements: tree level Symanzik improved gauge action;
stout smearing
lightest particles hard to measure: mesons with disconnected
contributions; glueballs
improvements: spectral decomposition, smearing techniques
12/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Low energy effective theory
confinement like in QCD ⇒ colorless low energy bound states
multiplet1 :
mesons : a − f0 : λ̄λ; a − η 0 : λ̄γ5 λ
fermionic gluino-glue (σµν Fµν λ)
multiplet2 :
glueballs: 0++ , 0−+
fermionic gluino-glue
Supersymmetry
All particles of a multiplet must
have the same mass
(scalar, pseudoscalar, fermion).
1
2
[Veneziano, Yankielowicz, Phys.Lett.B113 (1982)]
[Farrar, Gabadadze, Schwetz, Phys.Rev. D58 (1998)]
13/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
The gluino-glue particle
gluino-glue fermionic operator σ µν Tr[Fµν λ]
Fµν represented by clover plaquette
Lattice 24 × 48 κ = 0.1492
0.5
0.45
meff
0.4
0.35
0.3
0.25
0.2
2
4
6
8
10
12
14
t
⇒ APE smearing on gauge fields + Jacobi smearing on λ
14/18
Lattice SYM
Sign Problem
Summary of the Results
Mass gap at β = 1.6
Conclusions
1
⇒ unexpected mass gap
1
[Demmouche et al., Eur.Phys.J.C69 (2010)]
15/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
The influence of the finite lattice spacing
7
a−η ′
gluino-glue
6
r0 M
5
4
3
β
β
β
β
β
β
2
1
0
0
2
4
6
= 1.75
= 1.75
= 1.60
= 1.60
= 1.60
= 1.75
8
1 level stout
3 level stout
0 level stout
1 level stout
extrapolated
extrapolated
10
12
(r0 mπ )2
⇒ smaller lattice spacing considerably reduces the mass gap
16/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
The results of the mass spectrum: L = 1.35fm
7
gluino-glue
glueball 0++
6
r0 M
5
4
3
2
L = 1.8 fm, 1 level
L = 1.3 fm, 1 level
L = 1.8 fm, 3 level
L = 1.4 fm, 3 level
extrapolated value
1
0
0
2
4
6
8
stout
stout
stout
stout
10
12
(r0 mπ )2
still difficult to determine glueballs and a − f0
masses of the multiplet close to each other
17/18
Lattice SYM
Sign Problem
Summary of the Results
Conclusions
Conclusions and outlook
mass gap might be due to lattice artifacts
finite size effects: increase mass gap, but negligible in current
simulations
mass splitting is already hard to measure
at β = 1.75 on a 243 × 48 lattice
most important limitation: need large statistic,
especially for the scalar particles (0++ , a − f0 )
⇒ further improvements are investigated extended stout, clover
18/18